Non-Local Tug-of-War and the Infinity Fractional Laplacian

Details Non-Local Tug-of-War and the Infinity Fractional Laplacian Non-Local Tug-of-War and the Infinity Fractional Laplacian

2010-11-09

arxiv.org/abs/1011.1966v2

Mathematics

Analysis of PDEs

36 pages, 9 figures. Subsection 2.3 and the Appendix have been added to the second version

Scientific paper

Motivated by the "tug-of-war" game studied in [12], we consider a "non-local" version of the game which goes as follows: at every step two players pick respectively a direction and then, instead of flipping a coin in order to decide which direction to choose and then moving of a fixed amount $\epsilon>0$ (as is done in the classical case), it is a $s$-stable Levy process which chooses at the same time both the direction and the distance to travel. Starting from this game, we heuristically we derive a deterministic non-local integro-differential equation that we call "infinity fractional Laplacian". We study existence, uniqueness, and regularity, both for the Dirichlet problem and for a double obstacle problem, both problems having a natural interpretation as "tug-of-war" games.

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